AJP 81 (Dec. 2003): 569-572
Hudson on Receptacles
Timothy Bays
In a recent paper, Hud Hudson argues for what he calls the ‘liberal view of receptacles’ [Hudson 2002]. Here,
a ‘receptacle’ is a set of spatial points which could be exactly occupied by a material object, and the liberal
view is the view that any set of points constitutes a receptacle. Hudson’s argument for this conclusion is
complicated. For one thing, it’s structured indirectly. Hudson argues against three rival views of receptacles
(the view that material objects must occupy open regions, the view that they must occupy closed regions, and
the view that they must be three-dimensional); he then suggests that the liberal view ‘appears considerably
more attractive’ in the aftermath of these arguments. For another thing, the nature of the views Hudson is
arguing against leads him to engage in some complicated topological constructions.
Now, I have no desire to challenge Hudson’s arguments here. (Not being a metaphysician myself, I find
that I have no independent views on the nature of receptacles.) But I think it’s worth noting that the
mathematics in Hudson’s paper can be dramatically simplified—in particular, the topology can be almost
completely eliminated. Doing so will, I think, help to clarify the significance of Hudson’s arguments.
1
The Philosophy
Hudson’s arguments rest on three principles. The first two are highlighted near the beginning of his paper
[434–435]; the third comes out later in his argument [438].
(U) Necessarily, for any collection of material objects, the xs, there exists a material object, y,
such that the xs compose y.
(E) Necessarily, there are no spatially-extended material simples.1
(RP) Necessarily, for any material objects x and y, if x has y as a part, and y does not have x
as a part, then there exists a material object z such that x has z as a part and z does not
overlap y.
Along with these principles, Hudson assumes that space consists of a continuum of spatial points (so, it’s
neither granular nor gunky). He also alludes in passing to a ‘recombination principle’ which allows us to argue
directly from the existence of a one-point receptacle to the truth of the liberal view [436 and 438]. Without
1 It’s
not clear just what Hudson means here. He could mean simply that any extended material object has a proper part.
Or, he could mean that any extended material object is decomposable into proper parts—i.e., that there exist disjoint parts
hAi | i < αi such that the fusion of these parts gives us the original object. Judging from arguments on 435, 436 and 437, I
would guess that Hudson intends the latter principle. For our purposes, however, the earlier (weaker) principle will suffice.
Also, it’s not quite clear what Hudson means by ‘spatially-extended.’ In section 2, I assume that an object is ‘extended’ if its
receptacle contains more than one point. In section 3, I discuss the impact of other conceptions of ‘extension’ on my argument.
AJP 81 (Dec. 2003): 569-572
worrying about the exact nature of this principle,2 I’ll focus on a related principle which is supposedly a
consequence of Hudson’s principle—call it the ‘liberals win on points’ principle:
(LWP) If any point is a receptacle, then the liberal view is correct.
Together, these principles and assumptions provide the background for Hudson’s discussion.
In the next section, I will present a simple, non-topological, proof of the following result: (U), (E), and
(RP) entail that there are one-point receptacles. This result has three immediate consequences. First, it
provides a simple argument against two of Hudson’s rivals—the ‘open’ theory and the ‘three-dimensional’
theory—as neither of these two theories allows one-point receptacles. Second, it provides an argument against
the ‘closed’ theory, provided that this theory admits receptacles which have non-closed subsets.3 Finally, if
we go ahead and accept (LWP), then it gives a direct argument for Hudson’s liberal view. At the end of the
day, then, we get all of Hudson’s conclusions (and more!) with almost none of his work.
2
The Mathematics
Let A and B be material objects. For notational purposes, I will use Ā and B̄ to denote the receptacles
occupied by A and B, I will use A < B to abbreviate ‘A is a proper part of B,’ and I will use A ∪ B to
denote the fusion of A and B. I begin with a lemma:4
Lemma 1. Let A < B. Then Ā ( B̄.
Proof. Let A < B. By (RP), there exists A0 < B such that A and A0 do not overlap. This is incompatible
with Ā = B̄.
Q.E.D.
Theorem 2. Let p be a point in Ā. Then {p} is the receptacle for a part of A.
Proof. If {p} = Ā, then we are done. Otherwise, we construct an increasing sequence, A0 < A1 < A2 . . . , of
proper parts of A such that for each α, p 6∈ Āα . We begin by applying (E) to obtain B < A and then (RP)
to obtain B 0 < A such that B and B 0 do not overlap. If p ∈ B̄, we let A0 = B 0 ; otherwise we let A0 = B.
Given Aα , we apply (RP) to generate Bα < A such that Aα and Bα do not overlap. There are three
cases to consider:
2 Presumably,
Hudson’s recombination principle is supposed to do two things. First, it lets us ‘move receptacles around’ so
that, if one point is a receptacle, then any point is a receptacle. It also has some mechanism for ensuring that it’s possible for
any collection of receptacles to be jointly occupied (perhaps the ‘patchwork principle’ ?).
3 Two
remarks are in order. First, the result in the next section actually shows that any point within a receptacle is also a
receptacle. This, together with Hudson’s (U), is enough to generate the argument mentioned above. Second, the qualification
on ‘admitting receptacles with non-closed subsets’ is important. In principle someone could think that matter is only possible
at points with rational coordinates of the form ( 2m201 ,
m2 m3
,
)
220 220
where m1 . . . m3 are integers. This view is perfectly consistent
with (U), (E), and (RP), and it easily entails that all receptacles are closed.
4 In
the mathematics which follows, I make use of the Axiom of Choice (to justify transfinite induction up to (2ℵ0 )+ ). While
few mathematicians today have scruples about this axiom, it’s use in a metaphysical context should probably be noted. As far
as I can tell, Hudson’s own arguments do not invoke Choice.
AJP 81 (Dec. 2003): 569-572
Case 1. Suppose p 6∈ B̄α . Then we let Aα+1 = Aα ∪ Bα , noting that Aα+1 < A by (U).5
Case 2. Suppose p ∈ B̄α but {p} 6= B̄α . Then we apply (E) to obtain Cα < Bα and then (RP)
to obtain Cα0 < Bα such that Cα and Cα0 do not overlap. If p ∈ C̄α , we let Aα+1 = Aα ∪ Cα0 ;
otherwise we let Aα+1 = Aα ∪ Cα .
Case 3. Suppose {p} = B̄α . Then {p} is the receptacle for Bα and our theorem is done.
At limits, we let Aλ =
S
α<λ
Aα and note that Aλ < A by (U). By Lemma 1, the sequence of receptacles,
Ā0 ( Ā1 ( Ā2 . . . , is constantly increasing. For cardinality reasons, therefore, our construction must
terminate in less than (2ℵ0 )+ steps at some ‘Case 3’ situation. Eventually, therefore, we obtain a Bα < A
which has {p} as its receptacle.
3
Q.E.D.
A Caveat
Before concluding, I want to briefly mention a minor worry concerning the above argument. In applying
(E) in the proof of Theorem 2, I assumed that a region is ‘extended’ if it contains more than one point. It
is possible, however, that Hudson has a more restrictive conception of extension (perhaps, that a region is
extended iff it contains an open ball or iff it has non-zero topological dimension).
There are two things to say here. First, I think Hudson is committed to the ‘one-point’ version of (E).
Some of his arguments seem to presuppose that discontinuous objects have proper parts [435, 436 and 437];
others presuppose that continuous, non-point sized, objects have proper parts [435, 436 and 437]. Together,
these assumptions entail the ‘one-point’ version of (E). Second, and more importantly, a trivial modification
of the proof of Theorem 2 gives the following (for any conception of extension).
Theorem 20 . Let p be a point in Ā. Then there exists X ⊂ Ā such that p ∈ X and X is the receptacle for
a simple (hence, non-extended) material object.
On any reasonable conception of extension, this theorem is still enough to rule out the ‘open’ view and
the ‘three-dimensional’ view. Further, Hudson clearly assumes that the existence of non-extended closed
receptacles is incompatible with the ‘closed’ view [436]. If so, then the theorem rules out that view as well.
(And if not, then Hudson’s own arguments against the ‘closed’ view fail too.) At the end of the day, therefore,
we still get all of Hudson’s conclusions with almost none of his work (and, in particular, with virtually no
topology).
References
Hudson, Hud 2002. The Liberal View of Receptacles, Australasian Journal of Philosophy 80: 432–439.
5 Throughout
this paper, I interpret (U) as entailing that a fusion of parts of A is also a part of A (and not just a material
object whose receptacle is a subset of Ā). It is clear from, e.g., 438 that this is Hudson’s own understanding of (U).